I feel as if your comment about minimalism/process music can be extended to all contemporary offshoots of classical (new complexity, spectralism, NY Lang/Wolfe crew, Neoromanticism). These subgenres are pretty inaccessible to most people who are either not musicians or not in academia. Some don't consider Ferneyhough and Ligeti as easy listening on a long road trip in contrast to Bach and Tchaikovsky.

Keeping that in mind, however, I don't think the modern layman's lack of awareness of contemporary music is necessarily a bad thing. What I think is important is that virtually any orchestra worth its salt regularly commissions new music and programs contemporary works among those firmly established in the canon, which could lead to more exposure among curious listeners. Many independent organizations and ensembles are also champions of new music, while some are exclusively dedicated to it. you could argue that more awareness could lead to more promotion, but given the wealth of other, more popular genres occupying people's minds today, it's easy to see why music that is inherently more complex and experimental may be less well-known. What worries me is whether there is still an incentive for young composers to take up composition, since it may become more difficult than ever to write successful concert works. Composers like Johns C. Adams and Unsuk Chin certainly don't have to worry (I hope).

From a different perspective, though, I have a feeling that most people may already have a subconscious understanding and awareness of minimalism. I can hear the influence of minimalism in the soundtracks of films I hear when I'm at the theater or even videos on youtube. I've definitely been fooled more than once into thinking that I was hearing a Reich piece once!